Studies into Polyhedra - Prisms

Look around the Graphics Lab

Regular, Semi-Regular Polyhedra, and thier Duals (first page)
Prisms, Anti-prisms, Pryamids, and related Polyhedra
Miscellanous Polyhedra: Deltahedra
Johnson Solids -- The other convex polyhedra with regular faces

Why I studied polyhedra, and Image Generation Techniques
Known Polyhedral Mathematical Formula
Data Sources and links for Polyhedral Data

Prisms, Anti-prisms, Pyramids, and related Polyhedra

These sets consist of infinite series, and are generally generated using a standard formula from two polygons of the same type. These polygons do not even have to be regular or even convex polygons, though the anti-prism formula would require some regularity to the starting polygon.

Prisms

Take two polygons and connect together with vertical edges of the same length to form squares. Each vertex has 3 edges, the new edge being at right angles to the edges of the original polygonal interface.

The cylinder is a special 'infinite' case of this class.

This method is also known as elongation of the original polygons (See Johnson Solids).

triangular_prism off vrml	square_prism off vrml	pentagonal_prism off vrml
	hexagonal_prism off vrml	octagonal_prism off vrml	decagonal_prism off vrml

Note: A "Square Prism" is more commonly known as a "Cube".

Anti-prisms

Join the polygons with triangles. each vertex will then have 4 edges, contectin the upper polygon with its lower polygon which is twisted slightly.

The method of connecting is also known as gryo-elongation of the polygons. (See Johnson Solids).

triangular_antiprism off vrml	square_antiprism off vrml	pentagonal_antiprism off vrml
	hexagonal_antiprism off vrml	octagonal_antiprism off vrml	decagonal_antiprism off vrml

Note: A "Triangular Antiprism" is also the platonic "Octahedron".

Prism Duals

These may look like di-pyramids with unequal edge lengths but are much more specific. They are formed by taking the dual of a prism.

As the number of edges of the original polygon increases, the pyramids become very long pointy.

triangular_prism_dual off vrml	square_prism_dual off vrml	pentagonal_prism_dual off vrml	hexagonal_prism_dual off vrml
	heptagonal_prism_dual off vrml	octagonal_prism_dual off vrml	enneagonal_prism_dual off vrml	decagonal_prism_dual off vrml

The the "Square Prism Dual" is also a "Square Di-pyramid" which is a "Octahedron".

Anti-prism Duals

In a simular way these are the duals of a anti-prism. The faces are all kite shaped quadrilaterals, and like dual-prisms become very long as the number of faces increase.

triangular_antiprism_dual off vrml	square_antiprism_dual off vrml	pentagonal_antiprism_dual off vrml	hexagonal_antiprism_dual off vrml
heptagonal_antiprism_dual off vrml	octagonal_antiprism_dual off vrml	enneagonal_antiprism_dual off vrml	decagonal_antiprism_dual off vrml

The "Triangular Antiprism Dual" is also a "Cube".

ASIDE: the "Pentagonal Prism Dual" and "Pentagonal Antiprism Dual" are the only polyhedra that made a good ten-sided dice. The later is usually used as the points are better distributed, and are common in the paper and dice fantasy games such as advanced Dungons and Dragons.

Pyramids and Di-Pyramids

Basically forming a tent on a polygon. Note however that only three of these can be formed using edges of equal length, built using a triangle (forming a tetrahedron), a square (regular or square pyramid, like in Egypt), and a pentagon (or "pentagonal pyramid").

If you try this with a hexagon, the object is flat, with six triangles on top of the hexagon, and no volume. After that it's impossible, unless you use longer edges to form the pyramid, but then the object will not conform to any .

In other words, there are only three true pyramid polyhedra...

triangular_pyramid off vrml

square_pyramid off vrml

pentagonal_pyramid off vrml

Adding a pyramid to both sides of the original polygon will produce di-pryamids.

triangular_dipyramid off vrml

square_dipyramid off vrml

pentagonal_dipyramid off vrml

The "Triangular Pyramid" is a platonic "Tetrahedron", and the "Square Dipyramid" is a "Octohedron". The other four pyramidal solids form part the Johnson Solids Series (In sequence #1 #2 #12 and #13).

Aside: Note that the famous Egyptian Pyramid does NOT conform to a regular square pyramid, but has angles which is thought to be of astronomical relevence, rather than mathematically.

Cupolas, Bi-cupolas

These are gem like polygons which are formed by joining a polygon to another polygon with double the number of edges, using squares and triangles. Again only three such Cupolas exist.

triangular_cupola off vrml

square_cupola off vrml

pentagonal_cupola off vrml

Bicupolas are like pyramids, just two of the objects joined back to back. In each case however they can be joined together in two different ways, ortho-bi-cupola, are mirrored across the join, and gryo-bi-cupola have a small twist so two triangles are not connected together by an edge.

triangular_orthobicupola off vrml

square_orthobicupola off vrml

pentagonal_orthobicupola off vrml

triangular_gyrobicupola off vrml

square_gyrobicupola off vrml

pentagonal_gyrobicupola off vrml

The "Triangular Gyrobicupola" is also the Archimedian Solid "Cuboctahedron", so is the only object not part of the Johnson Solids.